Ordinal Analysis and Consistency Proofs

Session 1: 03/19, 17:00-18:00. Introduction, partial orders, wellorders, ordinal numbers.


Session 2: 04/09, 17:00-18:00.  Transfinite induction, ordinal arithmetic, Cantor's normal form theorem.


Session 3: 04/16, 17:00-18:00. The language of arithmetic, Robinson's arithmetic, Sigma_1-completeness, the pairing lemma.


Session 4: 04/23, 17:00-18:00. The arithmetical hierarchy, coding of finite sequences.


Session 5: 04/30, 17:00-18:00. Gödel numberings, provability predicates, the First Incompleteness Theorem, Rosser's Theorem.


Session 6: 05/14, 17:00-18:00. The Second Incompleteness Theorem, proof of the diagonal lemma.


Session 7: 05/21, 17:00-18:00. Turned out to be a holiday.


Session 8: 05/04, 17:00-18:30. Reflection principles, n-consistency, Beklemishev's Analysis of PA, I.


Session 9: 05/11, 17:00-18:30. Beklemishev's Analysis of PA, II.


Session 10: 05/25, 17:00-18:30. Afterword.



Some references:

Sessions 1,2: Jech, Set Theory.


Sessions 3–5,7: Hájek and Pudlák, Metamathematics of First-Order Arithmetic; Boolos, The Logic of Provability.


Sessions 8,9: Beklemishev, Provability Algebras and Proof-Theoretic Ordinals, I. Ann. Pure Appl. Logic. Joosten, Pi^0_1-Ordinal Analysis Beyond First-Order Arithmetic, Math. Commun.


Proof of the reduction property: Beklemishev, Proof-Theoretic Analysis by Iterated Reflection, Arch. Math. Logic.


Session 10 and beyond: Pohlers: Proof Theory: the First Step into Impredicativity; Simpson, Subsystems of Second-Order Arithmetic.